Volumes of Subset Minkowski Sums and the Lyusternik Region

We begin a systematic study of the region of possible values of the volumes of Minkowski subset sums of a collection of M compact sets in \(\mathbb {R}^d\), which we call the Lyusternik region, and make some first steps towards describing it. Our main result is that a fractional generalization of the Brunn–Minkowski–Lyusternik inequality conjectured by Bobkov et al. (in: Houdré et al. (eds) Concentration, functional inequalities and isoperimetry. Contemporary mathematics, American Mathematical Society, Providence, 2011) holds in dimension 1. Even though Fradelizi et al. (C R Acad Sci Paris Sér I Math 354(2):185–189, 2016) showed that it fails in general dimension, we show that a variant does hold in any dimension.